conjecture-making in higher arithmetic geometry
Andreas Holmstrom Uppsala University, Sweden andreas.holmstrom@gmail.com http://andreasholmstrom.org/
Abstract. We present a framework for encoding information about ob- jects from higher arithmetic geometry. This framework is built around a new kind of data type called a Tannakian symbol. The arithmetic objects we have in mind include modular forms (and more general automorphic representations), elliptic curves (and more general schemes, motives and algebraic stacks), finite graphs, group representations, and multiplicative functions (like the Euler totient function). The language of Tannakian symbols not only allows for representations of individual objects, but also representations of classes of objects, relations between objects, and vari- ous important unary and binary operations on objects. The development of this framework is the first small step in a long-term project aiming to apply machine-learning algorithms to some problems of current interest in modern arithmetic geometry.
Keywords: Tannakian categories, arithmetic geometry, zeta functions, motives, modular forms, lambda-rings, automated conjecture-making
1 Introduction
Arithmetic geometry is one of the most vibrant and abstract areas of modern pure mathematics. Out of the seven Millennium Problems, four come from pure mathematics, and of these four, one is solved and the remaining three belong to arithmetic geometry.
The prospect of artificially intelligent programs making new and deep dis- coveries in this area of mathematics is a tantalizing one. However, most of the concepts encountered in modern arithmetic geometry are not easily stored or manipulated by a computer. In the very long term, one may hope that advances in mathematical linguistics, as developed in Ganesalingam’s thesis [9], may lead to new computer-generated discoveries and proofs in arithmetic geometry. In the short term however, it is natural to look for other, less ambitious routes to making partial progress on selected problems.
In this project paper, we present a framework for encoding data about ob- jects from arithmetic geometry, with the aim of laying the foundation for future applications of machine-learning techniques in the field. We emphasize that this
is a first brief survey of a long-term project, focussing on examples. A more de- tailed discussion of potential applications will be provided in future publications and in discussions at the CICM conference, but as a first application, we pro- pose experiments with automated conjecture-making on invariants of motives and schemes, using the language of Tannakian symbols presented here, together with the SAGE package for domain-independent automated conjecture-making developed recently by Larson and Van Cleemput [14].
1.1 Arithmetic objects
Somewhat informally, we shall use the term “arithmetic object” to refer to any kind of object that is of central importance in arithmetic geometry. Some classes of such objects are: (1) Geometric objects (e.g. a scheme). The reader unfamiliar with the theory of schemes may think of a scheme simply as a system of polyno- mial equations. (2) Algebraic objects (a group, a ring, a Hopf algebra, etc.). (3) Homotopical objects (like an algebraic stack or a ring spectrum). (4) Combina- torial objects (for example a graph). (5) Analytic objects (e.g. a zeta function).
(6) Objects in a Tannakian category. Examples of the latter include represen- tations of finite groups, representations of Lie groups, Galois representations, automorphic representations, motives, Hodge structures, and F-isocrystals.
Technical definitions of all the above terms (schemes, Tannakian categories, etc) can be found in the online Encyclopedia of Mathematics [7]. Rather than giving all of these definitions here, we shall present explicit examples from most of these classes and explain how our proposed encoding framework applies to each example.
In addition to seeking encodings of single objects, a central goal of our work is to also encode information about classes of objects (for example the class of objects in some given Tannakian category), operations on objects (such as tensor product of group representations, or Tate twist of motives, or Dirichlet convolution of multiplicative functions), relations between objects (such as a representation being a direct summand of another), and invariants of objects (like the Euler characteristic of a scheme).
1.2 What would be required of a good encoding framework?
Let C be some class of arithmetic objects, for examples the class of all elliptic curves over the rational numbers, or the class of all finite undirected graphs, or the class of all complex representations of the Monster group.
We seek an encoding framework for objects in C satisfying the following properties:
1. To every objectX in the classCwe can assign a finite amount of structured dataE(X). (We think of E(X) as anelementary orelectronic ”shadow” of the objectX.)
2. Given a description of X, there should be an explicit algorithm computing E(X).
3. Many important invariants of X should be computable fromE(X) only.
4. Many important operations on objects in C should correspond to explicit manipulations of the corresponding structured data.
5. Given two objectsX and X0 from different classes (say one graph and one elliptic curve), the two associated pieces of data E(X) and E(X0) should
”be of a similar form” (to facilitate the discovery of connections between different kinds of structures).
6. Many of the deepest theorems and conjectures about objects X in modern arithmetic geometry should have a formulation in terms of the associated dataE(X) only.
Any encoding satisfying these requirements will have the property that a com- puter could in principle discover (or guess) interesting mathematical statements by searching for patterns in the structured data of many arithmetic objects.
We have found an approach that satisfies all of the above criteria for many classes of arithmetic objects. The framework is built around the notion of a Tannakian symbol. In many cases, the information contained in the Tannakian symbol is the same as the information contained in the “zeta function” of the arithmetic object, but Tannakian symbols are more flexible than zeta functions, and there are also cases where it makes sense to speak of Tannakian symbols even though there are no zeta functions around.
1.3 Previous work
We are not aware of any previous work with the explicit ambition of apply- ing machine-learning algorithms to geometric, Tannakian and homotopical cat- egories in higher arithmetic geometry. However, we have drawn inspiration from many places. Due to algorithmic breakthroughs over the past decade by Ked- laya [13], Harvey [11], [12], Costa and Tschinkel [5] and others, it is now possible to compute zeta functions of schemes in much higher dimensions and higher cohomological complexity than before, and these computations generate huge amounts of data, that can be interpreted in the language of Tannakian symbols.
A project with the aim of collecting this kind of data has been launched under the name the L-functions and Modular Forms Database (LMFDB) [15]. From another direction, we have been inspired by the now classical work of Zeilberger on holonomic sequences [20], the PhD thesis and articles of Colton [2], [3], [4] on automated conjecture-making in number theory, and of course the Online Ency- clopedia of Integer Sequences (OEIS) [17]. For a more comprehensive overview of previous work on automated conjecture-making, we refer to Larson and Van Cleemput [14].
2 Summary of algebraic theory
The aim of this section is to define what Tannakian symbols are, and to summa- rize their most important algebraic properties. Proofs of these statements will be given elsewhere.
2.1 Algebraic structures
We begin by recalling some definitions from abstract algebra. Amonoid is a set equipped with a binary operation that is associative and has an identity element.
A group is a monoid in which each element has an inverse. A monoid is called commutative if its binary operation is commutative. An abelian group is the same thing as a commutative group.
Example 1. The set of positive integersNis a monoid under addition, and it is also a monoid under multiplication. The set of all complex roots of unity is a monoid under multiplication.
A commutative ring is a set R with two binary operations, called addition (+) and multiplication (· ), with the requirements that R is an abelian group under addition, a commutative monoid under multiplication, and multiplication distributes over addition. The identity element for addition is denoted by 0, and the identity element for multiplication is denoted by 1.
Example 2. The set of integersZis a commutative ring. The setZ/m, identified with {0,1, . . . , m−1} is a commutative ring for any integer m ≥ 2, in which addition and multiplication are carried out modulo m. Whenever R is a com- mutative ring, the set R[x] of polynomials in xwith coefficients in R is also a commutative ring.
Afield is a commutative ring in which the nonzero elements under multipli- cation form a group (and not just a monoid).
Example 3. The set Qof rational numbers is a field, and so is the setRof real numbers, and the setCof complex numbers.
Amonoid homomorphismfrom one monoid to another monoid is a function which commutes with the binary operation and sends the identity element to the identity element. A ring homomorphism from a ring to another ring is a function which is a monoid homomorphism both with respect to addition and with respect to multiplication. An isomorphism (of rings or of monoids) is a homomorphism which admits a two-sided inverse.
Example 4. Letqbe a positive integer. It is known that there exists a field with exactly q elements if an only ifq is a power of a prime number (i.e.q=pe for some primepand some positive integere). Two such finite fields with the same number of elements are always isomorphic (i.e. there exists an ring isomorphism between them), and we writeFq for any finite field with exactlyqelements.
It is possible to describe all finite fields in a very concrete way. First of all, when q is a prime number, the ring Z/q is a field with q elements. A more interesting example is the field with four elementsF4, which can be described as the set{0,1, α, α+ 1}where addition is carried out modulo 2, and multiplication is carried out modulo 2and modulo the relationα2=α+ 1. Similar models of finite fields exist (but are not in general unique) for any prime powerq.
Alambda-ring is, informally, a commutative ringR“equipped with all pos- sible symmetric operations”. The precise definition of “all possible symmetric operations” is expressed in the notion of a lambda-structure on a commutative ring. The general definition of “lambda-structure” is given in terms of an infi- nite sequence λ0, λ1, λ2, . . . of functions (not ring homomorphisms!) from R to R, satisfying axioms that are a bit complicated. However, when the ringR is torsion-free (meaning that finite sums x+x+. . .+xare never zero unless xitself is zero), there is a simpler equivalent definition which we give here. All lambda-rings in this paper will be torsion-free, so this definition is enough for our purposes.
Definition 1. Let R be a torsion-free commutative ring. A lambda-structure on R is an infinite sequence of ring homomorphisms ψ1,ψ2, . . . from R toR satisfying the following axioms:
1. ψ1(x) =xfor allx∈R.
2. ψm(ψn(x)) =ψmn(x)for allm, nand all x∈R.
3. ψp(x)≡xp (modpR)for all prime numbers pand allx∈R.
The last condition means that the differenceψp(x)−xpcan be written as a mul- tiple ofp, in the ringR. The homomorphisms ψmare calledAdams operations.
2.2 Tannakian symbols
The kind of “structured data” we shall construct (denoted by E(X) in the in- troduction) will be called aU-indexedM-valued Tannakian symbol, and we now turn to the explanation of what this means.
Recall that amultisetis a unordered collection of elements, in which elements are allowed to be equal. For example,{2,2,2,5}is a multiset with four elements taken from the set of integers.
Definition 2. Let M be a monoid. An M-valued Tannakian symbol is an or- dered pair (A, B) of disjoint finite multisets with elements taken from M. We writeTS(M)for the set of allM-valued Tannakian symbols.
Conventions: We shall use the notationA/Bor AB for the ordered pair (A, B), and will refer toAas theupstairs multiset and toB as thedownstairs multiset.
Also, ifM happens to be a ring and we writeTS(M), we always think ofM as a multiplicativemonoid (in other words, we forget the additive structure).
Example 5. The symbol {1,1, i,−1,−i}/∅ is an example of a C-valued Tan- nakian symbol. Hereiis a complex square root of -1 and∅is the empty multiset.
Definition 3. Let U be a set. A U-indexed M-valued Tannakian symbol is a function fromU toTS(M). The set ofU-indexedM-valued Tannakian symbols will be denoted by TSU(M).
Example 6. Let P be the set of prime numbers, and let p denote a variable element ofP. Then{p2,1}/{p, p}is aP-indexedN-valued Tannakian symbol.
Consider multisetsA={a1, a2, . . .}, B ={b1, b2, . . .},C ={c1, c2, . . .} and D={d1, d2, . . .}where all the elements are taken from the same monoidM. We define operations on Tannakian symbols by the following formulas:
Addition: A
B ⊕C
D = A]C B]D (Here]denotes disjoint union of multisets.)
Multiplication: A B ⊗C
D = A·C]B·D A·D]B·C
(Here A·C denotes the monoid product of A and C, i.e. the multiset of all possible productsa·cwith a∈Aandc∈C, listed with repetition.)
Adams operations: ψnA B
={an |a∈A}
{bn |b∈B}
(Here, if the element a is repeated several times in A, the element an is also repeated the same number of times on the right hand side.) In each of these operations, it is understood that if the operation results in a symbol in which the upstairs and the downstairs multisets are not disjoint, then we remove pairs of identical elements until the multisets are disjoint. A few examples will illustrate what this means.
Example 7. Computations inTS(Z):
{5}
{1,−1}⊕{1,1,1}
{−1} = {5, 1,1,1}
{1,−1,−1} = {5,1,1}
{−1,−1}
{5}
{1,−1} ⊗ {10}
{3,7} = {50,3,7,−3,−7}
{15,35,10,−10}
ψ2{−1,−1,2,5}
{1,−2,7}
= {
(−1)2,(−1)2,22,52} {12,
(−2)2,72} ={1,25}
{49}
The main theorem about Tannakian symbols is the following:
Theorem 1. For any monoid M, the set TS(M) is a lambda-ring under the operations⊕,⊗andψn. The same is true forTSU(M)for any setU. Further- more,T SU(M)is functorial in M as well as in U.
One can go on and give explicit definitions of other structural features and invariants of Tannakian symbols, such as exterior powers, symmetric powers, virtual dimension, super-dimension, supertrace and superdeterminant. All this terminology comes from the setting of lambda-rings obtained by decategorify- ing Tannakian categories, but is retained also in situations where there is no Tannakian category involved. The point of all this structure is that whenever elements of the monoid M can be stored and manipulated by a computer, the same is true for elements of TSU(M), and the latter capture huge amounts of structure relevant for higher arithmetic geometry.
2.3 Assignments and fibers
Now we can be a bit more precise about the picture we would like to paint of structured data assigned to arithmetic objects. Return to the situation where C is some class of arithmetic objects. Given such a class, we can in many cases choose a monoidM and a setU and construct a map
E:C →TSU(M)
which satisfies most of the requirements in the introduction.
In such a setup, we are interested in the following general goals.
1. Understand how much information is lost when we pass fromX toE(X). A way of making this more precise is to define thefiber of a Tannakian symbol Sas the set of all arithmetic objectsX inC withE(X) =S. In many cases one can either prove that each fiber consists of at most one element, or give a bound on the size of the fiber.
2. Describe the image ofE.
3. Set up a correspondence between operations on arithmetic objects inC and operations on Tannakian symbols.
2.4 An elementary example: Linearly recursive sequences
Let a0, a1, a2, . . .be a linearly recursive sequence in C, with a0 = 1. It is well- known that it is then possible to rewrite the power seriesa0+a1t+a2t2+. . .as a rational expression of the formQn
j=1(1−βjt)/Qm
i=1(1−αit), and we define the Tannakian symbol attached to the linearly recursive sequence to be A/B, with the multisetsA={αi}andB ={βj}. With this definition, taking the product of power series corresponds to adding Tannakian symbols.
3 Schemes
3.1 General theory
For the purposes of this article, we define an affine scheme to be a finite set of variablesx1, x2, . . . , xd together with a finite list of polynomial equations (with integer coefficients) in these variables. Given an affine schemeX and a fieldK, we writeX(K) for the set of solutions to the equations ofX with values in the field K; elements of this set are calledK-valued points ofX.
We also define aprojective schemeto be a finite set of variablesx0, x1, . . . , xd
together with a finite list of homogeneous polynomial equations (with integer coefficients) in these variables. The equations are not required to be of the same degree. In this setting, we let X(K) denote the set of equivalence classes of solutions with values inK, where two solutions are called equivalent if one is a scalar multiple of the other. For projective schemes, we never count the trivial solution in which all variables take the value zero.
As a special case we may takeK to be the finite fieldFq, and the setX(Fq) is then automatically a finite set. We write #X(Fq) for the cardinality of this set.
There is a general construction which associates a projective scheme to any affine scheme. If the affine schemeX is defined by equations
fi(x1, x2, . . . , xd) = 0 i= 1,2, . . . m
then the associated projective scheme is defined by corresponding equations Fi(x0, x1, x2, . . . , xd) = 0 i= 1,2, . . . m
whereFi is obtained fromfi by multiplying each term by a suitable power ofx0 so thatFi becomes homogenous of degreedeg(fi).
Example 8. The equationx2+1 = 0 defines an affine schemeX (in one variable).
It is easy to see that in this case, we haveX(Q) = X(R) =∅ (the empty set), butX(C) ={i,−i}. Using modular arithmetic, we computeX(F2) =X(F3) =∅ andX(F5) ={2,3}. In general, for an odd primep, the cardinality ofX(Fp) is 2 or 0 depending on whetherpis congruent to 1 or 3 modulo 4. This pattern is a special case of Gauss’ famous quadratic reciprocity law, and quadratic reciprocity is an example of a pattern that can be expressed purely in terms of Tannakian symbols.
Theorem 2 (Dwork). Let X be a scheme (affine or projective) and let pbe a prime. There exists unique multisetsA={α1, α2, . . . , αm}andB={β1, . . . , βn} of complex numbers such that for allk≥1, we have
#X(Fpk) =β1k+β2k+. . .+βnk−αk1−αk2−. . .−αkm
Definition 4. LetX be a scheme and letpbe a prime. We define the Tannakian symbol ofX atpto beA/B, whereAandBare the multisets in Dwork’s theorem.
Example 9. Since Wiles proved Fermat’s Last Theorem using the Modularity theorem for elliptic curves, the class of elliptic curves has probably become the most famous class of schemes. As a simple example of an elliptic curve, take the scheme X defined by the equationy2+y =x3−x2. At the prime p = 2, the symbol becomes:
Affine case: {−1 +i,−1−i}
{2} Projective case: {−1 +i,−1−i}
{1,2}
The projective case is often the most interesting. In this example, deleting all numbers in the symbol except those with absolute value √
2 corresponds to cutting out the “motive” h1(X) from X. One can also associate Tannakian symbols to modular forms, and the Modularity theorem can be formulated as saying that for every elliptic curveX, there exists a modular form which at all primes has the same Tannakian symbol as the motiveh1(X).
Combining Tannakian symbols from all primes gives rise to a map from the class of elliptic curves to TSP(C). The fibers of this assignment are called isogeny classes of elliptic curves; it is known that these fibers are finite. Furthermore, elliptic curves come with a natural complexity measureN called the conductor (the above example has conductor 11), and by restricting attention to elliptic curves of, say, conductor less than 100000, we may restrict the set of indexing primes to a finite set without losing any information.
In general, the symbol attached to a projective scheme without singularities yields easy recipes for computing the Betti numbers and Euler characteristic of a scheme. In the above example the Betti numbers are 1, 2 and 1; these numbers are obtained by counting symbol elements with absolute value 1, √
2, and 2, respectively. The Euler characteristic is computed by subtracting the number of elements upstairs from the number of elements downstairs; for this elliptic curve we get 2−2 = 0. Plotting the numbers appearing in the symbol as points in the complex plane reveals symmetries related to Poincar´e duality and patterns related to the Riemann hypothesis over finite fields (proved by Deligne, Fields medal 1978). All of this was originally formulated as the famous Weil conjectures in the 1950s.
After computing the Tannakian symbols for several primes (up to size√ Nap- proximately) one can easily compute what’s called values of L-functions - these are the values appearing in the two Millennium Problems called the (global) Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture. These Tan- nakian symbols also allows for explicit formulations of many other deep questions of current interest to arithmetic geometers, such as the Sato-Tate conjecture, and various conjectures on Galois representations. The operations on Tannakian symbols correspond to operations in the so-called Grothendieck ring of motives, which is of interest not only in arithmetic geometry, but also in physics, where they are directly related to Feynman integral calculations in perturbative quan- tum field theory [16].
3.2 Case study: Arithmetic mirror symmetry
LetXκbe the ”quartic Dwork family”, i.e. the projective scheme defined by the equation
x4+y4+z4+w4= 4κxyzw
where κis a integer-valued parameter (so that by varyingκwe get afamily of schemes). The schemeXκ comes with a natural action of the groupZ/4×Z/4.
Taking the quotient scheme by this group action and resolving singularities yields a new schemeYκ, called the mirror ofXκ.
For concreteness, let’s look at the prime p = 41. For κ = 2, we get1 the following symbols:
E(X2) ={1,41,41,41,41,−41,−41, . . . ,−41,25−8√ 66i
2 ,25 + 8√ 66i
2 ,1681}/∅
1 The examples here are adapted from the presentation of Ursula Whitcher [19], and were computed using computer code by Edgar Costa [5].
Here there are 4 copies of the number 41 and 16 copies of the number -41. For the mirror variety, whicha priori might be expected to have a completely different symbol, we get
E(Y2) ={1,41,41, . . . ,41,−41,25−8√ 66i
2 ,25 + 8√ 66i
2 ,1681}/∅
with 19 copies of the number 41, a single copy of the number -41, and an other- wise identical symbol!
Still working withp= 41, for the caseκ= 3 we get:
E(X3) ={1,41,41, . . . ,41,−39 + 4√
10i,−39−4√
10i,1681}/∅
with 20 copies of the number 41. And this time, the Tannakian symbol for the mirror varietyY3 is
E(Y3) ={1,41,41, . . . ,41,−39 + 4√
10i,−39−4√
10i,1681}/∅
with 20 copies of 41, which means. . . that the symbols are absolutely identical!!
Patterns of this kind is the subject of arithmetic mirror symmetry, a relatively recent field inspired by the physics of string theory. It is conceivable that a computer searching for patterns in Tannakian symbols could have identified the schemes Xκ and Yκ as ”similar”, even if no human had ever thought of mirror symmetry.
4 More examples
4.1 Multiplicative functions
Much of elementary number theory (questions about primes, divisibility, etc.), can be formulated in terms ofmultiplicative functions fromNtoC. In this con- text a functionf is multiplicative iff(1) = 1 and f(mn) =f(m)f(n) whenever mandnare coprime.
Let p be a prime. For all multiplicative functions appearing naturally in number theory, it turns out that the sequence of function values
f(1), f(p), f(p2), f(p3), . . .
is linearly recursive, and hence we can associate a Tannakian symbol to the pair (f, p). Letting p vary over the set P of all prime numbers, we get a P- indexed C-valued Tannakian symbol attached to the multiplicative functionf. For example, the Euler totient function has symbol {p}/{1}, the characteristic function of the square numbers has symbol{1,−1}/∅, and the sum-of-divisors function has symbol{1, p}/∅.
This assignment is injective on multiplicative functions, and for many clas- sical classes of functions it stays injective even whenU is reduced to a finite set of primes. Furthermore, Dirichlet convolution of functions correspond to addi- tion of symbols, product of function corresponds to product of symbols (under a certain hypothesis), and norm operators on multiplicative functions correspond to certain Adams operations.
4.2 Graphs
There are at least three interesting ways of associating a Tannakian symbol to a (finite) graph. Firstly, given a graphX, we could define the Tannakian symbol of X to beA/∅, whereAis the spectrum ofX, i.e. the multiset of eigenvalues of the adjacency matrix ofX. With this definition, taking the disjoint union of graphs would correspond to addition of Tannakian symbols, and taking tensor product of graphs would correspond to multiplication of Tannakian symbols. Graphs with the same spectrum are calledisospectral, so the fibers of this assignment would be classes of isospectral graphs.
Example 10. With this definition, the Tannakian symbol of the complete graph on 4 vertices would be{−1,−1,−1,3}/∅
It is conjectured that almost all graphs are determined by their spectra. How- ever, there are many cases of non-isomorphic graphs with identical spectrum.
For example, the number of simple graphs on 9 vertices is 274668 (see OEIS:
Sequence A000088), while the number of such graphs isospectral to at least one other graph is 51039 (OEIS: Sequence A099883).
A second approach would be to define the Tannakian symbol of a graphX to be A/∅, where A={α1, α2, . . . , αm} is the finite multiset of complex numbers appearing in the expression
ζX(T) = 1
(1−α1T)(1−α2T)· · ·(1−αmT) whereζX(T) is the Ihara zeta function of the graphX.
Example 11. With this alternative definition, the Tannakian symbol of the com- plete graph on 4 vertices would be
{−1,−1,1,1,1,2,−1+
√7i 2 ,−1+
√7i 2 ,−1+
√7i 2 ,−1−
√7i 2 ,−1−
√7i 2 ,−1−
√7i 2 }/∅
In a recent paper [6], Durfee and Martin conjecture that almost all graphs which are not determined by their spectrum are determined by their zeta function.
As a third possibility, one can associate a certain polynomial (the “graph polynomial”) to any graphX. This polynomial defines a scheme, called the graph hypersurface ofX, and we could associate Tannakian symbols to the graphXby counting points of its graph hypersurface, like we did in the previous section for an elliptic curve and the quartic Dwork family. We refer to Brown and Schnetz [1]
for background, definitions, and extensive calculations motivated by applications to quantum field theory. One of their conclusions can be reformulated by saying that for many graphs, the Tannakian symbols constructed by this method seem to also come from modular forms.
4.3 Representations of finite groups
Let G be a finite group. Associated to G is its representation ring R(G), a lambda-ring generated by irreducible complex representations under the oper- ations of direct sum, tensor product, and exterior powers. Any element g ∈G
gives rise to a lambda-ring homomorphism from R(G) toTS(M), whereM is the monoid of complex roots of unity. Combining several such maps, one obtains a lambda-ring homomorphismEfromR(G) intoTSU(M), whereU is a subset of G. The most interesting choice ofU, which we will use in the remainder of this section, is to pick one representative of each conjugacy class ofG; this choice guarantees the injectivity ofE.
Many interesting patterns and unsolved problems about representations can be reformulated in terms of the map E:R(G)→TSU(M). This is due to the fact that both the character table ofGand the lambda-ring structure of R(G) can be recovered from the values of E.
Example 12. Taking G to be the Monster group, the character table is a 194 by 194 matrix, whose rank is 163. The number 163 also appears in the study of imaginary quadratic number fields; it is in fact the largest possible integerD such that the number fieldQ(√
−D) has class number 1 (meaning that its ring of integers is a unique factorization domain). The study of such number fields goes back to Gauss and is the simplest instance of Gauss’ famous class number problem. As explained for example in the popular book of Mark Ronan [18], the appearance of the number 163 in both places might well be a coincidence, but it could also be a hint that there is some mysterious connection between the Monster group and algebraic number theory that is yet to be understood.
An even more spectacular pattern connected with the Monster group is the Monstrous Moonshine Conjecture, formulated by Conway and Norton and proved by Richard Borcherds (Fields medal 1998). The starting point of this wonderful story was the observation that a certain number obtained from the character table (the dimension of the smallest nontrivial irreducible represen- tation) is (almost) equal to the coefficient of the linear term in the Fourier expansion of Klein’sj-function.
Let’s now turn to a much simpler group, for which everything can be worked out by hand.
Example 13. One of the simplest non-trivial examples of a finite group is the symmetric groupS3of permutations on three objects. This group has 6 elements in total, partitioned into 3 conjugacy classes. Letebe the identity element, let t be any transposition (an element of order 2), and let r be one of the two
“rotations” (an element of order 3). These three elements represent the three conjugacy classes of S3. The number of irreducible representations of a finite group is the same as the number of conjugacy classes, and in our example, the irreducible representations are:
C+: The trivial representation, sending every permutation to 1.
C−: The sign representation, sending a permutation to its sign.
C2: A two-dimensional representation, visualized as a matrix action of S3 on a triangle with vertices at the three cube roots of unity in the complex plane.
In this simple case, it is easy to compute the functionE by hand. We get, for the three different choices of group elementg:
Caseg=e: E(C+) ={1}/∅ E(C−) ={1}/∅ E(C2) ={1,1}/∅
Caseg=t: E(C+) ={1}/∅ E(C−) ={−1}/∅ E(C2) ={1,−1}/∅
Caseg=r: E(C+) ={1}/∅ E(C−) ={1}/∅ E(C2) ={ω, ω2}/∅
Hereωis a primitive 3rd root of unity. Any element ofR(G) can be written as a formal differenceV−W, whereV andW are representations built as direct sums of irreducible ones, and we may compute in R(G) by identifying such a formal difference with an ordered triple of symbols (using the injective map E and applying the rules for computing with Tannakian symbols). For example, we get (suppressing E from the notation): C2 = {1,1}/∅ , {1,−1}/∅ , {ω, ω2}/∅
and C+ ⊕C− −C2 = ∅/∅ , ∅/∅ , {1,1}/{ω, ω2}
. A similar computation shows thatC2⊗C2 equals C+⊕C−⊕C2, and in general any tensor product of representations can be expressed as a direct sum of irreducibles, using only Tannakian symbols.
4.4 Algebraic stacks
The arithmetic objects discussed so far in this paper were key players in many of the greatest arithmetic discoveries of the 20th century. However, in 21st century research, new classes of objects are becoming increasingly important, and these objects come from homotopy theory and higher category theory. The most beau- tiful application so far is probably the recent proof of the Tamagawa number conjecture by Gaitsgory and Lurie [8], in which homotopical objects calledstacks play a prominent role. Stacks are a generalization of schemes, for which X(Fq) is no longer a set, but agroupoid or asimplicial set. There are several different ways of assigning Tannakian symbols to (certain classes of) stacks. The method used for schemes will work provided we accept infinite multisets. For example, the stackBGm (the classifying stack of the multiplicative group) would then at the primephave the Tannakian symbol{p−1, p−2, p−3, . . .}/∅.
5 Final remarks on arithmetic pattern-detection
What constitutes an important discovery or a deep conjecture in arithmetic geometry? Looking at many examples in the literature, a few of which we have seen in the present survey, it is reasonable to say thatdeep arithmetic statements are often observations of patterns that can be expressed in terms of Tannakian symbols.
But is it conceivable that automated algorithms really could have detected some of these patterns? And is it reasonable to expect machines to discover new patterns, maybe even of comparable interest to the Weil conjectures, Monstrous Moonshine, or arithmetic mirror symmetry? Although we do not claim to know the answer to these questions, we would like to end by suggesting two necessary
features that such algorithms would have to incorporate in order to have any chance of making such discoveries.
5.1 Exotic metrics
In traditional data analysis, data points are given by vectors of real numbers (or rather floating point approximations), visualized as points in n-dimensional Euclidean spaceRn. Two data points are then considered to have similar features if they are close with respect to some metric on Rn derived from the standard metric (x, y)7→ |x−y| on the set of real numbers.
Similarly, in a traditional neural network based on perceptrons or sigmoid neurons, the output of an individual neuron is a function of the size of an in- coming real number, and “size” here refers to a measurement made using the standard metric on the real numbers.
In arithmetic geometry, many patterns and relations can be expressed in terms of a metric, but it is not enough to work with the standard metric on real numbers. Instead, the standard metric needs to be complemented by others, most importantly thep-adic metrics. For any prime numberp, thep-adic metric on the set of rational numbers is defined as follows. For two distinct rational numbersxandy, there is a unique integerksuch thatx−y can be written on the form±pk·a/b, whereaandb are positive integers coprime top. Thep-adic distance |x−y|p betweenxand y is defined to bep−k. This definition can be extended to irrational algebraic numbers, but unlike the standard metric, it is not defined for transcendental numbers.
Unravelling the definition, it is easy to see that as a special case, two integers xandyare congruent modulopif and only if they are close in the sense that their p-adic distance is less than or equal to 1/2. Combining different primes and using the Chinese remainder theorem, any arithmetic pattern involving congruences can be expressed (and hence potentially discovered) usingp-adic metrics2. 5.2 Symmetry detection
Another class of arithmetic patterns can be collected under the umbrella of sym- metry. While the lambda-ring structure on Tannakian symbols in itself captures certain kinds of symmetry, there are also many other kinds, including Poincar´e duality (seen in the symbol of a projective scheme without singularities), various symmetries in Hodge diamonds, symmetries of modular forms, and the fact that the number of rows in a character table equals the number of columns.
A mathematical treatment of symmetry invariably involves group theory, and in situations where the symmetry group is both finite and known, it is easy to implement algorithms for detecting symmetry. However, in cases where the symmetry group is infinite and/or unknown, the symmetry detection problem
2 It might also be interesting to build algorithms based on other kinds of metrics, like theI-adic metric on a polynomial ring or a Grothendieck ring (whereI is an ideal of the ring), or the Granville-Soundararajan metric on multiplicative functions [10].
is more challenging. It would be interesting to explore the image recognition and computer vision literature on symmetry detection and think about whether known algorithms can be applied to arithmetic settings, for example to plots of complex numbers taken from a Tannakian symbol.
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